Fourier Theory without Complex Analysis

Abstract:

One of the reasons why the Fourier transformation is a useful tool is that it converts differentiation (hard to understand) to multiplication (easier to understand). Many of the basic proofs of Fourier theory use complex analysis.

This is a beautiful theory, but unstable under perturbations (for instance, to work on manifolds), as it relies on exact computations that cannot be carried out when things change a little. 

In recent years, there has been interest in avoiding the use of complex analysis.

The talk explains some of the progress and the difficulties.